Reasoning and proof Knowledge-Based Practice

Questions

1. 1.

   Which property of equality justifies the conditional statement: “If $AB + BC = AC$ and $BC = DE$, then $AB + DE = AC$”?

   • A) Symmetric Property of Equality
   • B) Substitution Property of Equality
   • C) Transitive Property of Equality
   • D) Reflexive Property of Equality
2. 2.

   Which of the following figures serves as a valid counterexample to the conjecture: "If a quadrilateral has four congruent sides, then it must have four right angles"?

   • A) A rectangle with adjacent side lengths of 4 and 6
   • B) An isosceles trapezoid with non-parallel congruent legs
   • C) A rhombus with interior angles measuring $60^\circ$ and $120^\circ$
   • D) A square with all side lengths equal to 5 units
3. 3.

   Given the true conditional statement “If two lines in a plane are perpendicular to the same line, then they are parallel” and the fact that distinct lines $p$ and $q$ in plane $M$ are both perpendicular to line $r$, which form of deductive reasoning directly validates the conclusion that $p \parallel q$?

   • A) Law of Syllogism
   • B) Inductive Generalization
   • C) Law of Contraposition
   • D) Law of Detachment
4. 4.

   Which of the following is the contrapositive of the conditional statement: "If two angles are vertical angles, then they are congruent"?

   • A) If two angles are congruent, then they are vertical angles.
   • B) If two angles are not congruent, then they are not vertical angles.
   • C) If two angles are not vertical angles, then they are not congruent.
   • D) Two angles are congruent if and only if they are vertical angles.
5. 5.

   Given the true conditional statements: "If a quadrilateral is a rhombus, then its diagonals are perpendicular" and "If the diagonals of a quadrilateral are perpendicular, then they divide the quadrilateral into four right triangles." Which conclusion follows directly by the Law of Syllogism?

   • A) If the diagonals of a quadrilateral divide it into four right triangles, then the quadrilateral is a rhombus.
   • B) If a quadrilateral is a rhombus, then its diagonals divide the quadrilateral into four right triangles.
   • C) If a quadrilateral is not a rhombus, then its diagonals do not divide it into four right triangles.
   • D) If the diagonals of a quadrilateral are perpendicular, then the quadrilateral is a rhombus.

Answers and step-by-step solutions appear after you finish the test.